Infinite Descent

The above equation has no integer solutions

Please teach me how to prove the above using the infinite descent

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Yes Its quite complicatedSee the last step .... it claims that they have reached infinite descentHow?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Yes.. that one

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Hi;

To have an infinite descent in this case you have to prove that there is always integers like the ones you started with that have the same properties but are smaller. For any finite a,b you can not keep finding integers p,q that are smaller than a,b. You must run out.

Re: Infinite Descent

But now we have reached infinite descent since:P^2 = a^2 + b^2 = p which is less than p^2 + q^2 = z which is less than z^2.

But how does this show that a and b will have similar properties as p & q??

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

So what?They are squares... how does that show that there will be smaller numbers MATCHING THE ORIGINAL CONDITION?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Thats hard to believe(Just joking)

Last edited by Agnishom (2012-09-13 02:51:23)

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Pythagorean triangle?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Okay, imagining the two legs are perfect squares, say a^2 and b^2We can't have a hypotenuse c such that a^4 + b^4 = c^2

But what if we are talking about one leg and one hypotenuse?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

I do not understand the question... To have relatively prime numbers, you need two numbers.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Infinite Descent

The theorem claims that there exists integers x and y such that and Example, 10^2 = 25*4 where (25, 4) are relatively prime to each other now see that 25 = 5^2 and 4 = 2^2

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

The theorem claims that there exists integers x and y such that and Example, 10^2 = 25*4 where (25, 4) are relatively prime to each other now see that 25 = 5^2 and 4 = 2^2

If , take and . Otherwise, since and are coprime, each prime divisor of must divide exactly as many times as it divides , namely a multiple of times. Thus , being a product of primes of power a multiple of , is a power of . Same for .

Re: Infinite Descent

Are you sure the proof is as simple as that?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: Infinite Descent

Re: Infinite Descent

I still have confusions, though it looks intutively obvious

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember